Current Variations

The power generation in SOFCs via oxidizing hydrogen with oxygen flowing in the re-spective flow channels is illustrated in Fig. 1.5. For an infinitesimal current density ∂i

Figure 1.5: The power generation principle of SOFCs by using hydrogen and oxygen flowing in the regarding flow channels.

generated within an area of ∂x (for the unit width) depicted in Fig. 1.5, the consumed

reactant fluxes can be written as

∂J_H^∗

2 = ∂i

n_H2F ∂J_O^∗

2 = ∂i

n_O2F (1.33)

according to the Faraday’s law in the anode and cathode, respectively. Here,

∂i= ∂I

∂x (1.34)

Since the consumed reactants are converted to the water-vapor

− ∂J_H^∗

2

∂t = ∂J_H^∗

2O

∂t =−1 2

∂J_O^∗

2

∂t (1.35)

the concentrations of the reactants and product vary as

− ∂c^∗_H

2

∂t = ∂c^∗_H

2O

∂t =−1 2

∂c^∗_O

2

∂t (1.36)

Regarding the continuous flow of the reactants and product along the respective flow channels within the electrodes, the longitudinal concentration variation can be expressed as

− ∂c^∗_H2

∂x = ∂c^∗_H2O

∂x =−1 2

∂c^∗_O2

∂x (1.37)

As stated by Eq. 1.23, for a constant η_a comprising the concentration and the activation polarizations in the anode, current is a function of the reactant and product concentrations.

Because the concentrations vary along the channel, Eq. 1.23 becomes i(x) = i₀

c^∗_H

2(x) c^b_H

2(x)exp

βnF η_a RT

−c^∗_H

2O(x) c^b_H

2O(x)exp

−(1−β)nF η_a RT

(1.38) Since the drops in the reactant concentrations give rise to the product concentration,

i(x) = i₀ c^∗_H

2(x)−^∂c_∂x^∗H2 c^b_H

2(x)−^∂c_∂x^∗H2 exp

βnF η_a RT

− c^∗_H

2O(x) + ^∂c_∂x^∗H2O c^b_H

2O(x) + ^∂c_∂x^∗H2O exp

−(1−β)nF η_a RT

(1.39) current decreases along the channel. This longitudinal decrease in the current is equivalent to the reversible voltage-loss referred to as “Nernst-loss” [28,29].

1.3.2 Temperature Variations

While converting the chemical energy of hydrogen into power through the HOR at current densityi, heat ˙q_tot(W/m²) is unfavorably released due to the entropy change Δs(J/mol K) and Nernst-loss, together categorized as the reversible polarizationη_rev; and due to the acti-vation, ohmic, and concentration polarizations, categorized as the irreversible polarization η_irev, as given in the following equation

˙

q_tot =i(η_rev +η_irev) (1.40)

At a given polarization η_tot, which includes both the reversible and irreversible contribu-tions, current varies along the channel owing to the concentration variacontribu-tions, assuming that the variations in the other directions are negligible. As a result, the heat production rate becomes a function of the position

˙

q_tot(x) =i(x)η_tot (1.41)

implying that temperature is also a function of position along the channel.

On the other hand, the released heat is removed through the conductive, radiative, and convective heat transfer processes to maintain the cell temperature at a desired level. The conductive heat transfer along the x axis can be defined according to the Fourier’s law as

˙

q_cond(x) =−k∂T_s

∂x (1.42)

Due to the large temperature difference between the cell and ambient temperature, radiant heat transfer plays an important role. According to the Stefan-Boltzman law, the maximum rate of radiation emitted from a black body at the position of x

˙

q_emit,max(x) =σ_SBT_s⁴(x) (1.43)

where σ_SB is the Stefan-Boltzman constant. In fact, the emissivity of a real object diverges from the unity (0 ≤ ≤ 1) that adopted for the black body. This divergence modifies Eq. 1.43 as

˙

q_emit(x) = σ_SBT_s⁴(x) (1.44)

Additionally, the absorptivity of a real object α departs from that of the black body (0≤α≤1). As a result, the absorbed heat from the surrounding is defined as

˙

q_abs(x) =α q˙_inc(x) (1.45)

where ˙q_inc (W/m²) denotes the incident radiation.

In this case, the net radiant heat transfer at the position of xcan be expressed as follows

˙

q_rad(x) = ˙q_abs(x)−q˙_emit(x) (1.46) Owing to the continuous flow of the reactant and product gases, the convective heat transfer is also effective for removing the heat from cell. Depending on the heat transfer coefficient h (W/m²K) and the inlet gas temperatureT_∞, the convective heat transfer at the position of x can be written according to the Newton’s law as

˙

q_conv(x) =h(T_s(x)−T_∞) (1.47) Depending on the balance between the heat production rate and the total heat transfer rate, consisting of the conductive, convective, and radiant heat transfer rates, the surface temperature differentiates along the channel

T_s =f(x) (1.48)

Herein the surface temperature refers to either of the anode or cathode. Thanks to the rather small thickness and high thermal conductivity (k (W/m²K)) of the components (subsection 1.2.2), the temperature difference among the components in the through-plane direction (y-direction in Fig. 1.5) is numerically estimated to be trivial.

Endothermic cooling

Despite hydrogen is considered in the previous analyses, the tolerance of SOFCs to the hydrocarbon fuels is one of their advantages. As depicted in Fig. 1.6, there are two driving forces for sticking on the hydrocarbon fuels. Firstly, the infrastructure for storing and distributing hydrogen is yet insufficient, whereas the storage and distribution of hydrocar-bons is relatively easy. In fact, methane (in the city gas) is effectively distributed all over

Figure 1.6: The concept of direct internal reforming in SOFCs.

the globe. Secondly, at the moment, the most of hydrogen (ca. 96%) is produced from hydrocarbons via various processes [30]; thus, the direct use of hydrocarbons theoretically is more efficient owing to the reduced entropy loses [31]. Thanks to these convincing facts, various hydrocarbon fuels, such as methane, ethane, and butane, etc. are under investiga-tion to utilize for power generainvestiga-tion in SOFCs. Among them, methane has been regarded as the most promising fuel.

Nickel being the most commonly used catalyst of SOFCs is very active for methane cracking reaction

CH₄ C+ 2H₂ (1.49)

which leads to “carbon deposition”, resulting in irreversible degradations, as discussed in subsection1.3.3. Therefore, the electrochemical oxidation of methane has been achieved for only short-term. Similarly, the “dry methane reforming” often terminates the cell operation with the carbon deposition. Fortunately, methane has being successfully reformed via the

endothermic MSR (methane steam reforming) reaction

CH₄+ 2H₂O CO₂+ 4H₂ ΔH =−195.8 (kJ/mol) (1.50) As shown in Fig. 1.6, hydrogen is usually produced through the MSR in the catalytic steam reformers, where nickel is employed as the catalyst as well. The operation temperature of the catalytic reactors approaches to the typical operation temperature of SOFCs. These analogies have been inspiring for the unification of the MSR and the HOR in SOFCs and referred to as “direct internal reforming”. Although, the original idea is to exploit the water-vapor and heat produced by the exothermic HOR for the endothermic MSR, unfortunately this idea can only partially be realized due to the carbon deposition. Thereby, methane is fed along with steam at a particular S/C (steam/carbon) ratio to prevent the carbon deposition.

Depending on the concentration of the species, the rates of the HOR and the MSR reaction change along the channel. It is often reported that the MSR reaction rate is rather high in the fuel inlet vicinity and decreases longitudinally. Depending on the rate of the MSR reaction, the SOFC is cooled down at longitudinally varying degrees via the “endothermic cooling”. This type of cooling is expected to be rather effective according to the numerical calculations.